Parallel Guided Local Search and Some Preliminary Experimental Results for Continuous Optimization

David C. Wyld et al. (Eds) : CCSIT, SIPP, AISC, PDCTA, NLP - 2014

pp. 421–425, 2014. © CS & IT-CSCP 2014 DOI : 10.5121/csit.2014.4236

Nasser Tairan

, Muhammad Asif Jan

and Rashida Adeeb Khanum

1

College of Computer Science, King Khalid University, Abha, KSA

[email protected]

2

Department of Mathematics, Kohat University of Science & Technology, Kohat 26000, KPK, Pakistan

[email protected]

3

Jinnah College for Women, University of Peshawar, Peshawar, KPK, Pakistan

[email protected]

A

This paper proposes a Parallel Guided Local Search (PGLS) framework for continuous optimization. In PGLS, several guided local search (GLS) procedures (agents) are run for solving the optimization problem. The agents exchange information for speeding up the search. For example, the information exchanged could be knowledge about the landscape obtained by the agents. The proposed algorithm is applied to continuous optimization problems. The preliminary experimental results show that the algorithm is very promising.

K

Guided Local Search, Continuous Optimization Problems, Parallel Algorithms, Cooperative algorithms

1.

I

Many real-world applications can be modeled as optimization problems [1][2]. These problems may have many local optima and/or have no analytical form available. Traditional mathematical programming methods are not suitable for dealing with them. During the last two decades, many heuristic methods have been developed. Many of them such as simulated annealing [3], tabu search[4], and guided local search [5] are single point iteration based methods. The research work in the Evolutionary Computation and Particle Swarm Intelligence has demonstrated that a population-based search scheme often has some advantages over single-point based methods on efficiently utilizing computer memory and parallel computing process units. Therefore, a natural issue arises: how can a single point based heuristic be populationized? Mistunori and Ogura [6], E-Abd and Kamel[7]and Blum and Roli [8]have made some attempts on this issue.

422 Computer Science & Information Technology (CS & IT)

information, learned from the previous search about the problem, with each other periodically for speeding up the search. Some preliminary experiments have been carried out to study the effectiveness of the proposed PGLS up the search.

The rest of the paper is organized as follows: the proposed PGLS is described in Section 2. Sections 3 and 4 list the test instances and report experimental results. Section 5 concludes this paper.

2.

P

G

L

S

2.1. Guided Local Search

Guided Local Search (GLS) is a very general strategy for guiding a local search algorithm to escape from local optima. In GLS, solution features are defined to distinguish between solutions with different characteristics, so that bad characteristics can be penalized and hopefully removed. The solution features depend on the problem and the local search. Features can be defined in a very natural way. For example, in the travelling salesman problem (TSP), a feature can be a link between two cities. When the search is trapped in a local optimum, GLS will use the following function as the objective function:

h s = g s + λ p × I s , (1)

where sis a candidate solution and g(s) is the original objective to minimize, is a control parameter, I ranges over the features. pis the penalty for feature i. Ii(s) indicates whether or

not solution s exhibits feature i:

=

otherwise. 0

; feature exhibits

if 1 )

(s s i

I_i

All thep_iare initialized to be 0. When the local search traps at a local optimums*, GLS computes the utility function of penalizing feature i:

util s∗ _{= I s}∗ _{× (2)}

Where c is the cost for the feature i (in TSP, the cost for a feature is the distance for that

feature), p is the current penalty value forfeature. GLS penalizes the feature with the highest

utility value, i.e., increase its p value by 1. The LS will then continue with the new augmented objective function.

2.2. PGLS

In PGLS, several agents run GLS in a parallel way and exchange information periodically for speeding up the search. Different agents start their searches from different points in the solution

space. PGLS (manager) records the best solution

x

consideration by these agents. After every K penalizations, each agent conducts a crossover

from this new solution. The p values for each agent will be reset to zero. The algorithm with n agents works as follows:

Step1: Initialization

Generate randomly n points x₁, ,x_n in the search space. Step2: Guided Local Search

2.1 For i=1 to n

Agent i does GLS from _! until K penalizations have been made and returns its best solution "_!found so farand its current solution#_!. 2.2 Set " to be the best solution to the problem among all the"_!’s. Step 3

3.1 If the stopping condition is met, return ". 3.2 Otherwise,

3.2.1 For i=1 to n, do crossover on " and #_! to produce a new solution

!.

3.2.2 Go to Step 2.

2.2.1. Local Search

Following Voudris [9], each real-valued variable in a continuous optimization problem is represented by a number of bits. Therefore, the flip bit mutation is used as a local search method. The local search moves from one solution to another by flipping the value of a bit in the solution. It starts from a random bit and then examines all the possible bits.

2.2.2. Crossover

In the crossover, 2n crossover sites are first randomly selected under the constraint that there are two crossover sites in each variable substring, and then 2 child solutions are generated by swapping along these sites. The child solution with lower original objective function value is returned as a new solution (Step 3.2.1).

3.

E

S

We have tested the above PGLS algorithm to minimize Rastriging function, Rosenbrock function, and Schwefel function. These three functions are:

f$%= 10n + ) *x,-− 10 cos 2πx, 2 3 4

,5

f6 4= 0, 8x,= 09, −5.12 ≤ x,≤ 5.12

f$== ) >100 x, − x,- -+ x,− 1 -? 4 4A

,5

f6 4= 0, 8x,= 19, −2.048 ≤ x,≤ 2.048

fCD= 418.9829n − ) Fx,sinGHx,HI 5 4

424 Computer Science & Information Technology (CS & IT)

f6 4= 0, 8x,= 19, −500 ≤ x,≤ 500

In our experiments, each real-valued variable is represented in 22 bits as in [9]. Therefore, each solution is represented in 22n bits when n is the number of variables.

The GLS used in our experiments is the same as in [9]. The number of agents=2. The value of K varies based on the problem as it is shown in Table 1. The algorithm stops after each agent has done 100× K penalizations.

3.

R

The obtained results show that the proposed cooperative method, PGLS, was the most effective algorithm of the two algorithms. It showed a good performance and produced good results for the lunched functions in every run. Table 1 shows the experimental results for the run algorithms

Table 1. Comparison between the run algorithms

4.

C

REFERENCES

[1] R. Horst, N.V. Thoai and P Kluwer Academic Publishers, [2] P.M. Pardalos and H. E. Ro Publishers, Boston, MA, 2002. [3] Kirkpatrick, S., C. D. Gelatt Jr

4598, 671-680, 1983. [4] Fred Glover , Fred Laguna, Ta [5] Voudouris, C, Guided local se of Computer Science, Universi [6] Hiroyasu, T., Miki, M. and

Proceedings of the IASTED Systems, Las Vegas, pp. 145-1 [7] El-Abd, M. and Kamel, M., heuristics (Springer Berlin - Le [8] Blum, C. and Roli, A. “Meta Comparison”. ACM Computin [9] Voudouris, C., Guided Loca Technology Journal, Vol.16, N

AUTHORS

Nasser Tairan received B.Sc. degre M.Sc. degree in Software Engineerin Science (AI) from University of Ess currently Assistance Professor in K research areas are evolutionary comp Muhammad Asif Jan received the Pehawar, Khyber Paktunkhwa, Pak University of Essex, UK, in 2011. H of Mathematics, Kohat Uiversity Paktunkhwa, Pakistan. His main unconstrained/constrained single/mu algorithms, decomposition based objective optimization, mathematica

Rashida Adeeb Khanum received t Pakistan in 1997and the Ph.D. from in 2012.

P.M. Pardalos, Introduction to Global Optimization, Se s, Boston, MA, 2000.

omeijn (Eds.), Handbook of Global Optimization, Kluw 2.

Jr., M. P. Vecchi, "Optimization by Simulated Annealing", Tabu Search , Kluwer Academic Publishers, Norwell, MA, search for combinatorial optimization problems, PhD Thes rsity of Essex, Colchester, UK, July, 1997.

d Ogura M., “Parallel simulated annealing using genet D International Conference on Parallel and Distribute

150, 2000.

., “A Taxonomy of Cooperative Search Algorithms”. Lectures in Computer science), pp. 32-41, 2005.

etaheuristics in Combinatorial Optimization: Overview an ting Surveys, vol. 35, no. 3, pp. 268- 308, 2003.

cal Search -- An illustrative example in function opt , No.3, July 1998, 46-50.

gree in computer science from King Abdulaziz University, ring from University of Bradford, UK in 2005, M.Sc. degre ssex, UK in 2007 and P.hD. degree from University of Ess King Khalid University, College of Computer Science, K

mputation, single and multiobjective optimization and meta he M.Sc. degree in mathematics from University of akistan in 1997 and the Ph.D. in mathematics from . He is currently the Chairperson of the Department ity of Science & Technology (KUST), Khyber in research areas are evolutionary computation, multi- objective optimization by using evolutionary d evolutionary methods for constrained multi-cal programming, and numerimulti-cal analysis.

d the M.Sc. degree from University of Peshawar, m University of Essex, Colchester, United Kingdom

Second Edition, luwer Academic

",Science, 220, , 1997.

esis, Department etic crossover”, uted Computing . Hybrid and Conceptual

ptimisation, BT

y, KSA in 1999, ree in Computer ssex 2012. He is KSA. His main